Analyzing large and high-dimensional flow data sets is a non-trivial task and favorably carried out using sophisticated tools which allow to concentrate on the most relevant information and to automate the analysis. These goals can be achieved using topological methods, which foster target-oriented studies of the most important flow features. They have a variety of applications which ranges from a skeletal representation of the overall flow behavior to a detailed analysis of vortex structures. This thesis presents novel algorithms and approaches for the extraction, tracking and visualization of topological structures of vector fields. The new concept of connectors is introduced which allows visually simplified representations of topological skeletons of complex 3D vector fields. The first visualization technique for 3D higher order critical points and the corresponding classification are presented. Based on this theory, two novel applications for the topological simplification and construction of 3D vector fields have been developed. Furthermore, the first generic approach to feature extraction is presented, which allows to extract and track a rich variety of geometrically defined, local and global features evolving in scalar and vector fields. The use of generic concepts and grid independent algorithms aims at a broad applicability of the extraction methods while alleviating the implementational expenses. Further contributions include the first topology-based visualization approach for two-parameter-dependent 2D vector fields and a thorough study of vortex structures. The usefulness of the newly developed methods is shown by applying them to analyze a number of data sets. The work presented in this thesis has been published in peer-reviewed international conference proceedings, journals, and books.